Problem: Simplify the following expression: $y = \dfrac{-5x^2+16x+16}{x - 4}$
Answer: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(16)} &=& -80 \\ {a} + {b} &=& &=& {16} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-80$ and add them together. Remember, since $-80$ is negative, one of the factors must be negative. The factors that add up to ${16}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-4}$ and ${b}$ is ${20}$ $ \begin{eqnarray} {ab} &=& ({-4})({20}) &=& -80 \\ {a} + {b} &=& {-4} + {20} &=& 16 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-5}x^2 {-4}x) + ({20}x +{16}) $ Factor out the common factors: $ x(-5x - 4) - 4(-5x - 4)$ Now factor out $(-5x - 4)$ $ (-5x - 4)(x - 4)$ The original expression can therefore be written: $ \dfrac{(-5x - 4)(x - 4)}{x - 4}$ We are dividing by $x - 4$ , so $x - 4 \neq 0$ Therefore, $x \neq 4$ This leaves us with $-5x - 4; x \neq 4$.